## Algebraic Topology, Geometry and Physics

### Past events

#### Spring 2015

May 12, 2015
Yannick Voglaire (University of Luxembourg)
Differentiable stacks from the point of view of Lie groupoids - Part 2
In this second talk, I will continue to describe the relation between differentiable stacks and Lie groupoids, focusing this time on Lie groupoids. I will provide explicit examples and, if time permits, I will explain the relation to C*-algebras.
April 21, 2015
Alessandro Zampini
How to define a Dirac operator following Kähler's approach
April 7, 2015
Stephen Kwok
On theories of superalgebras of differentiable functions, 5
We will conclude our discussion of the work of Carchedi and Roytenberg on the foundations of super Fermat theories, the natural superalgebraic generalizations of Fermat theories. In this talk, we will continue our discussion of super Fermat theories and show that every ordinary Fermat theory may be extended canonically to a super Fermat theory.
March 31, 2015
François Petit (University of Luxembourg)
Stable and presentable infinity-categories
The theory of presentable and especially presentable and stable ∞-categories provides a very powerful framework. For instance, stable presentable categories form a monoidal category, there is also an adjoint functor theorem and a version of Brown representability theorem for such categories. In this talk, I will present, following Lurie, a brief overview of the theory of stable and presentable categories from the viewpoint of the user. If time permits, I will also discuss spectra and stabilization.
March 24, 2015
Alessandro Zampini
Spin geometry and Dirac operators, part 2
Monday, March 23, 2015, from 4:00 to 5:30 PM
Alessandro Zampini
Spin geometry and Dirac operators, part 1
March 17, 2015
Yannick Voglaire (University of Luxembourg)
Differentiable stacks from the point of view of Lie groupoids
In this introductory talk, I will introduce differentiable stacks and show how they are presented by Lie groupoids. I will describe generalized morphisms between Lie groupoids and the resulting notion of Morita equivalence. I will then explain in which precise (bicategorical) way isomorphic differentiable stack correspond to Morita equivalent Lie groupoids.
March 10, 2015
Stephen Kwok
On theories of superalgebras of differentiable functions, 4
We will continue our discussion of the work of Carchedi and Roytenberg on the foundations of super Fermat theories, the natural superalgebraic generalizations of Fermat theories. In this talk, we will discuss nilpotent extensions in the context of Fermat theories, superalgebras, and super Fermat theories.
March 3, 2015
Stephen Kwok
On theories of superalgebras of differentiable functions, 3
We will continue our discussion of the work of Carchedi and Roytenberg on the foundations of super Fermat theories, the natural superalgebraic generalizations of Fermat theories. In this talk, we will continue to discuss the development of differential calculus in the context the Fermat theories of Dubuc-Kock.
February 17, 2015
Stephen Kwok
On theories of superalgebras of differentiable functions, 2
We will continue our discussion of the work of Carchedi and Roytenberg on the foundations of super Fermat theories, the natural superalgebraic generalizations of Fermat theories. In this talk, we will present further background material on the Fermat theories of Dubuc-Kock, and the development of differential calculus in the context of such theories.
Thursday, January 29, 2015, from 2:15 to 3:45 PM
Stephen Kwok
On theories of superalgebras of differentiable functions, 1
Dubuc and Kock developed the concept of Fermat theories, theories of commutative algebras for which infinitely differentiable functions may be evaluated on elements. These theories arose in the development of models for synthetic differential geometry, but have more recently played a key role in the development of models for derived differential geometry.
In this series of talks, we will discuss the work of Carchedi and Roytenberg on the foundations of super Fermat theories, the natural superalgebraic generalizations of Fermat theories. In this first talk, we will present background material on Lawvere algebraic theories, and the Fermat theories of Dubuc-Kock.
January 16-17, 2015
Informal meetings on the occasion of the visit of Luca Vitagliano (University of Salerno)

#### Fall 2014

December 2, 2014
Damjan Pištalo
The BRST construction, Part II
This is the second part of last week's talk. We will combine the Koszul-Tate and Longitudinal differentials to construct the BRST complex.
November 25, 2014
Damjan Pištalo
The BRST construction, Part I
In my last talk, I defined the longitudinal differential. I will now show that its zero homology is equal to the space of observables. I will then define the Koszul-Tate resolution as a semifree resolution of $C^\infty(\Sigma)$, where $\Sigma$ is the constraint surface. Combining Koszul-Tate and Longitudinal differential, I will construct the BRST complex.
November 18, 2014
Damjan Pištalo
Constrained Hamiltonian systems, III
This is the third talk in our series. After reviewing the symplectic structure on the phase space induced by the Poisson bracket, we will show that zero vector fields of the induced two form on the constraint surface are generators of infinitesimal gauge transformations. Moreover, such vector fields satisfy Frobenius' theorem on the constraint surface. This allows us to define the de Rham complex mentioned previously.
November 11, 2014
Damjan Pištalo
Constrained Hamiltonian systems, II
This is the second part of last week's talk.
November 4, 2014
Damjan Pištalo
Constrained Hamiltonian systems, I
In gauge theory, equations of motion contain arbitrary functions of time. This leads to relations (constraints) between the canonical variables $p$ and $q$ in the Hamiltonian formalism, restricting the motion to a submanifold of the phase space (the constraint surface). Constraints are divided into first and second class. The existence of first class constraints corresponds to the existence of gauge symmetries, which is the case when more than one set of canonical variables describe the same physical state. Vector fields generating infinitesimal gauge transformations satisfy Frobenius' theorem on the constraint surface. This enables us to define something similar to de Rham complex over their “duals”. At the end, I will establish a 1-1 correspondence between these vector fields and first class constraints, that will be crucial for the antighost-ghost correspondence needed in the BRST formalism.